Problem 27 - Precalculus : Building Concepts and Connections

Question

Find a possible expression for a quadratic function \(f(x)\) having the given zeros. There can be more than one correct answer. $$x=1 \text { and } x=-3$$

Step-by-Step Solution

Answer

A possible expression for the quadratic function is \(f(x) = x^2 + 2x - 3\).

Step 1: Identify the Roots

The roots of the function are given as \(x = 1\) and \(x = -3\). We'll denote these as \(r\) and \(s\) respectively.

Step 2: Formulate the Quadratic Function

The quadratic function with roots \(r\) and \(s\) is \(f(x) = a(x - r)(x - s)\), where \(a\) is a non-zero constant. So we have \(f(x) = a(x - 1)(x + 3)\). Given there can be multiple correct answers due to different values of \(a\), for simplicity's sake, let's take \(a = 1\). Therefore, our function becomes \(f(x) = (x - 1)(x + 3)\).

Step 3: Simplify the Function

The next step is to simplify the function. Multiply out the brackets to yield a standard quadratic function: \(f(x) = x^2 + 2x - 3\).

Key Concepts

Zeros of a Function

In mathematics, the zeros of a function are the values of the variable that make the function equal to zero. When we talk about a quadratic function \(f(x)\), these zeros are essentially the points where the graph of the function crosses the x-axis. Consider a quadratic function like \(f(x) = ax^2 + bx + c\). The zeros of this function are the solutions to the equation \(f(x) = 0\).

To find the zeros, we often look for values of \(x\) that satisfy the equation. For example, if given zeros are \(x = 1\) and \(x = -3\), they mean that at these points, the function evaluates to zero, i.e., \(f(1) = 0\) and \(f(-3) = 0\). Therefore, the expression for the function can be expressed using these zeros, ultimately helping us in constructing the quadratic equation itself.

Roots of Quadratic Equations

The roots of a quadratic equation are another way to refer to the zeros of a quadratic function \(f(x) = ax^2 + bx + c\). Solving for these roots is essential in understanding where the function has no output, i.e., when \(f(x) = 0\).

Using the quadratic formula, it's possible to find these roots:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

For simpler situations, such as when the equation is already factored, the roots can be immediately found. In our specific example, with zeros at \(x = 1\) and \(x = -3\), our quadratic expression takes the form \(f(x) = (x - 1)(x + 3)\).

Expanding this expression gives us the standard quadratic form, making it easier to see the relationship between the quadratic function and its graph.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents. Quadratic functions are a type of polynomial expression, specifically of degree two, as indicated by their highest exponent.

For a quadratic polynomial such as \(f(x) = x^2 + 2x - 3\), it includes:

• The quadratic term \(x^2\)
• The linear term \(2x\)
• The constant term \(-3\)

To build this polynomial from given zeros, we start from its factored form \((x - 1)(x + 3)\). Expanding it results in \(x^2 + 3x - x - 3\), simplifying to \(x^2 + 2x - 3\). This illustrates how zeros define the factor structure of a polynomial, directing us towards its standard form through multiplication.